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Next: Acknowledgements and Further Reading Up: Projections of Cubic Crystals Previous: 3. Cubic Crystals

4. The Diagrams

In each computer drawn diagram, the Miller indices of the face(s) defining the form(s) are shown under the letters H, K, L (printed by the computer as capital letters). The distance of this face from the centre, on an arbitrary scale, is shown under D. The point group symmetry (shown at the top of each diagram under the word CUBIC) then operates on the specified face (or faces) to give the complete solid. For example, m 3 m symmetry generates all six faces of the cube from the single face (100). For the solids thus generated from a single face, four orthographic views are given with hidden edges omitted: a `general' view along the [621] axis, closely similar to a standard clinographic projection; and views along the [100], [110] and [111] directions. For each cubic crystal viewed along the [111] direction the three-fold symmetry is evident. There are three other equivalent three-fold axes along [1$\overline{11}$], [$\overline{1}$1$\overline{1}$] and [$\overline{11}$1]. One can also see immediately whether the crystal possesses a four-fold axis along [100], a diad axis along [110], or mirror planes bisecting any of the projections along [100], [110], and [111].

As far as possible, similarly shaped solids are placed together in the pages which follow. In some cases two different settings (`positive' and `negative') are given for the same crystal: (3 & 4, 9 & 10, 15 & 16, 19 & 20, 25 & 26); and some crystals are mirror images (enantiomorphs) of one another: (22 & 23, 27 & 28).

Table 2 lists the names of the crystal forms and table 3 shows their distribution amongst the five cubic point groups, or crystal classes.

The cube (no. 1) appears in all five columns because any one of the cubic point group symmetries operating on the (100) plane will generate all six faces of the cube. The rhombic dodecahedron (5) also appears five times for the same reason; whilst the octahedron (2) appears three times, and the tetrahedron (3) twice. $\{$210$\}$, $\{$310$\}$ and $\{$320$\}$ are particular examples of $\{h k
0\}$; $\{$211$\}$ and $\{$311$\}$ of $\{$hll$\}$ with h > l ; and $\{$221$\}$ is an example of $\{$h h l$\}$ with h > l . $\{$321$\}$ is a particular case of the general form $\{hkl\}$ in which all the indices are different and non-zero. In table 3, it will be seen, for example, that the same crystal form (15) will be generated either by $\overline{4}$ 3 m or by 23 operating on (211).

Very often crystals exhibit faces of more than one form together. Some examples of combinations of two forms are shown here. The overall shape of a crystal is very much determined by the relative distances from the centre of the two kinds of face: one dominant form perhaps being only slightly modified by another. The drawings are still of ideal crystals, in which all faces of each form have the same size and shape. Real crystals seldom have the faces of forms so uniformly developed.

Graphical index


 
Table 2: Cubic crystal forms: key to the figure numbers
Fig. No. Form (Maximum) Symmetry Name No. of faces
1 $\{$100$\}$ m 3 m Cube 6
2 $\{$111$\}$ m 3 m Octahedron 8
3 $\{$111$\}$ $\=4 3 m$ Tetrahedron (positive) 4
4 $\{1\=11\}$ $\=4 3 m$ Tetrahedron (negative) 4
5 $\{$110$\}$ m 3 m Rhombic dodecahedron 12
6 $\{$210$\}$ m 3 m Tetrahexahedra 24
7 $\{$310$\}$
8 $\{$320$\}$
9 $\{$210$\}$ m 3 Pentagonal dodecahedra (or pyritohedra) 12
10 $\{$120$\}$
11 $\{$310$\}$
12 $\{$320$\}$
13 $\{$211$\}$ m 3 m Icositetrahedra (or trapezohedra) 24
14 $\{$311$\}$ m 3 m    
15 $\{$211$\}$ $\=4 3 m$ Tristetrahedra 12
16 $\{2\=11\}$
17 $\{$311$\}$
18 $\{$221$\}$ m 3 m Trisoctahedron 24
19 $\{$221$\}$ $\=4 3 m$ Deltoid dodecahedron (or deltohedron) 12
20 $\{2\=21\}$ $\=4 3 m$    
21 $\{$321$\}$ m 3 m Hexoctahedron 48
22 $\{$321$\}$ 4 3 Pentagonal icositetrahedra (or gyroids)  
23 $\{$312$\}$ 4 3   24
24 $\{$321$\}$ $\=4 3 m$ Hexatetrahedron 24
25 $\{$321$\}$ m 3 Didodecahedron (or diploid)  
26 $\{$312$\}$ m 3   24
27 $\{$321$\}$ 2 3 Tetrahedral pentagonal dodecahedra  
28 $\{$312$\}$ 2 3 (or tetartoids) 12
(The figures continue unnumbered for the combinations of forms.)
(29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40)



 
Table 3: Distribution of crystal forms amongst the five cubic classes
  m 3 m 4 3 $\=4 3 m$ m 3 2 3
$\{$100$\}$ 1 1 1 1 1
$\{$110$\}$ 5 5 5 5 5
$\{$111$\}$ 2 2 3 2 3
$\{$hk0$\}$ 6 6 6 9 9
$\{$hll$\}$ h > l 13 13 15 13 15
$\{$hhl$\}$ h > l 18 18 19 18 19
$\{$h k l$\}$ 21 22 24 25 27
The numbers are figure numbers. See Table 2 for the names of the forms.


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Next: Acknowledgements and Further Reading Up: Projections of Cubic Crystals Previous: 3. Cubic Crystals

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